You have a horizontal grindstone (a disk) that is 85 kg, has a 0.31 m radius, is turning at 88 rpm (in the positive direction), and you press a steel axe against the edge with a force of 21 N in the radial direction.

a) assuming the kinetic coefficient of friction between steel and stone is .2, calculate the angular acceleration of the grindstone in rad/s^2

b) how many turns will the stone make before coming to rest

Explanation:

moment of inertia of wheel

I = 1/2 mr² , m is mass and r is radius of disk .

= .5 x 85 x .31²

= 4.08425  kg m²

88rpm

=  1.467 x 2π rad /s

= 9.21 rad / s

force of friction acting = 21 x .2

= 4.2 N

torque produced by friction to stop it

= friction x radius

= 4.2 x .31

= 1.302 Nm

angular deceleration = Torque / moment of inertia

= 1.302 / 4.08425

= .31878 rad /s²

angular deceleration = -  .31878 rad /s²

b )

initial angular speed ω₁ = 9.21

Final angular speed ω₂  = 0

angle of turn = θ radian

ω₂² = ω₁ ² - 2 αθ , α is angular deceleration

0 = 9.21² - 2 x .31878 θ

θ = 133.04 radian

no of turns = 133.04 / 2π

= 21.18 turns .

root word, prefixes, and suffixes

russian is not my first language, so i'm afraid i can't write out an answer in russian, but hopefully you can find a way to do so. here's my best translation of the problem: you're given a plot of a particle's position in one-dimensional space over time, and you're asked to find either the total distance traveled by the particle after 6 seconds had elapsed, or the total displacement of the particle after 6 seconds.

if you want the total distance traveled, you would first need to determine the particle's velocity as a function of time, then integrate that over the 6-second interval. we know the position function is quadratic with roots at 0 and 6, so we have

where is unknown. because the position is parabolic, there is symmetry about the midpoint at , and we know that

so that the position function is

which means the velocity is

then the total distance traveled is

so the total distance traveled is 20 m.

on the other hand, if you want to find displacement: the particle starts at and ends at , so its displacement is 0.